Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C13×C16 · Schur multiplier (H2) = C23×C4 |
| direct product | C13×#13189 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8×C13 | C23 |
| commutator | C22 | C22×C13×C16 |
| Frattini, cosocle | C22×C8 | C23×C13 |
| socle | C23×C13 | C22×C8 |
| prime radical | C22×D4×C13 | C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C8×C13 ◁ G | |
534 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 1664 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(15,17)(19,21)(22,23,25,29,26,30,33,36,27,24,28,31,34,32,35,37);(14,15,16,17)(22,24,25,31,26,32,33,37,27,23,28,29,34,30,35,36);(18,19,20,21)1456 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 6177);